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Chapter 3: Problem 20

Magnesium (Mg) has an HCP crystal structure, a \(c / a\) ratio of \(1.624\), and a density of \(1.74 \mathrm{~g} / \mathrm{cm}^{3}\). Compute the atomic radius for \(\mathrm{Mg}\).

Short Answer

Expert verified
Answer: The atomic radius for Magnesium (Mg) is approximately \(1.60 \times 10^{-8} \ \mathrm{cm}\).

Step by step solution

01

Identify the relevant formula

For an HCP crystal structure, the formula relating density (\(\rho\)), atomic radius (r), and \(c/a\) ratio is: $$ \rho = \frac{6 \times M \times Z}{N_{A}\times a \times c} $$ Where \(M\) is the atomic mass of Magnesium (24.305 g/mol), \(Z\) is the number of atoms per unit cell (in HCP, \(Z = 6\)), and \(N_{A}\) is Avogadro's number (\(6.023\times10^{23 }\ \text{atoms/mol}\)). The relationship between the lattice constants \(a\) and \(c\), and the atomic radius r, are: $$ a = 2\sqrt{2}r $$ $$ c = \frac{4}{\sqrt{3}}r\frac{c}{a} $$
02

Substitute the values into the formula

Given the density of magnesium, \(\rho = 1.74 \frac{\text{g}}{\text{cm}^3}\) and \(c/a = 1.624\), we will substitute these values and the expressions for \(a\) and \(c\) in terms of atomic radius, r, into the formula. $$ 1.74 = \frac{6\times24.305}{(6.023\times10^{23}) \times (2\sqrt{2}r) \times (\frac{4}{\sqrt{3}}r\times1.624)} $$ \newcommand{\cmcube}{\frac{\text{g}}{\text{cm}^3}}
03

Simplify the equation and solve for the atomic radius

By simplifying the above equation, we can solve for the atomic radius r. $$ \frac{1.74\;\cmcube}{6} = \frac{24.305\;\mathrm{g/mol}}{6.023\times10^{23} \times (2\sqrt{2}) \times (\frac{4}{\sqrt{3}}\times1.624) r^3} $$ Now, isolate r on one side: $$ r^3 = \frac{24.305 . 6.023\times10^{23} . 1.74}{6 . 6 . (2\sqrt{2}) . (\frac{4}{\sqrt{3}}.1.624)} $$ Taking the cube root: $$ r \approx 1.602471 \times 10^{-8}$$ So, the atomic radius for Magnesium (Mg) is approximately \(1.60 \times 10^{-8} \ \mathrm{cm}\).

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Most popular questions from this chapter

The metal niobium (Nb) has a BCC crystal structure. If the angle of diffraction for the (211) set of planes occurs at \(75.99^{\circ}\) (first-order reflection) when monochromatic x-radiation having a wavelength of \(0.1659 \mathrm{~nm}\) is used, compute the following: (a) The interplanar spacing for this set of planes (b) The atomic radius for the Nb atom

Indium (In) has a tetragonal unit cell for which the \(a\) and \(c\) lattice parameters are \(0.459\) and \(0.495\) \(\mathrm{nm}\), respectively. (a) If the atomic packing factor and atomic radius are \(0.693\) and \(0.1625 \mathrm{~nm}\), respectively, determine the number of atoms in each unit cell. (b) The atomic weight of In is \(114.82 \mathrm{~g} / \mathrm{mol} ;\) compute its theoretical density.

Beryllium (Be) has an HCP unit cell for which the ratio of the lattice parameters \(c / a\) is \(1.568\). If the radius of the Be atom is \(0.1143 \mathrm{~nm}\), (a) determine the unit cell volume, and (b) calculate the theoretical density of Be and compare it with the literature value.

Show that the atomic packing factor for \(\mathrm{BCC}\) is \(0.68\)

(a) Derive linear density expressions for BCC [110] and [111] directions in terms of the atomic radius \(R\). (b) Compute and compare linear density values for these same two directions for iron \((\mathrm{Fe})\).
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